Trace of an Operator

Let $H$ be a finite-dimensional complex Hilbert space and let $A:H\to H$ be a linear operator. The trace of $A$, denoted $\operatorname{Tr}(A)$, is defined by choosing any orthonormal basis $(e_1,\dots,e_n)$ and setting

This value is independent of the chosen orthonormal basis.

This notion agrees with the usual matrix trace (see Trace ) after identifying $A$ with its matrix in an orthonormal basis.

Equivalent descriptions (finite dimension)

• If $A$ is represented by a matrix $(A_{ij})$ in any basis, then $\operatorname{Tr}(A)=\sum_i A_{ii}$.
• $\operatorname{Tr}(A)$ equals the sum of eigenvalues of $A$, counted with algebraic multiplicity.

Key properties

For operators $A,B$ on $H$ and scalars $\alpha,\beta$:

• Linearity: $\operatorname{Tr}(\alpha A+\beta B)=\alpha\operatorname{Tr}(A)+\beta\operatorname{Tr}(B)$.
• Cyclic property: $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$.
• Unitary invariance: if $U$ is unitary, then $\operatorname{Tr}(U^\ast A U)=\operatorname{Tr}(A)$.
• Positivity: if $A$ is positive semidefinite, then $\operatorname{Tr}(A)\ge 0$.

Trace and quantum expectation values

If $\rho$ is a Density Operator and $A$ is an observable (self-adjoint operator), the expected value of $A$ in state $\rho$ is

In particular, density operators are normalized by the condition $\operatorname{Tr}(\rho)=1$.

Note on infinite dimension

On an infinite-dimensional Hilbert space, not every bounded operator has a trace. One typically restricts to trace-class operators to extend $\operatorname{Tr}$ with similar properties.